The solution of partial differential equations (PDEs) related to flow phenomena is of great importance in a variety of application areas such as physics and engineering. However, the use of numerical methods to solve these equations often presents significant challenges. These include, for example, the need to discretize the problem, the choice of appropriate boundary conditions, and the computational cost of the simulation.

This seminar-style course is intended for students who have prior knowledge of discretizing PDEs using finite element or finite difference methods. The goal of the course is to provide an introduction to these techniques and their applications in fluid dynamics. The course covers a wide range of models and methods in fluid dynamics, including those related to laminar and turbulent flows, multiphase and interfacial flows.

Overall, the course aims to equip students with the skills and knowledge required to tackle challenging problems in fluid dynamics using numerical methods. Students will work on projects that will be presented at the end of the semester.

Resources

Online:

https://cloud.wias-berlin.de/aotearoa/index.php/s/ywt4GTBnqbKrPjx
Password: "numflu23navier"

Here you find videos and my handwritten notes and the topics for the seminar presentation. Also go to the "Resources" section of the whiteboard to find additional material.

with now online as video lectures:

Lecture 01: Introduction (April 19)
Lecture 02: 1.1 Methods of characteristics and Riemann problem (April 26)
Lecture 03: 1.2 Finite Differences in 1D (May 3)
Due to the discussion May 3rd, in Lecture 04 we will discuss finite differences in 2D.
Lecture 04: 1.2 Finite Differences in 2D (May 10)
Lecture 05: 1.3 Finite Elements (May 17)
Lecture 06: 1.4 Advanced Finite Elements (May 24)
Lecture 07: NEW: Introduction to FEniCS via Colab (May 30)
Lecture 08: 2.1-2.3 Stokes and Navier-Stokes

with practical demonstrations on

• Convection-diffusion
• Parabolic problems
• Constraints
• First steps Chorin projection
• Visualization via Paraview

Codes:

For the sources codes we decided to use Google Colab for the Course

https://colab.research.google.com

and I will make Python codes available for this course using a Github repository. You can directly import the codes into Colab from my repository "dpeschka/NumFluids" or clone from "https://github.com/dpeschka/NumFluids.git".

PDEs:

• Braess. Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics (Cambridge)
• Grossmann, Roos, Styles. Numerical Treatment of Partial Differential Equations (Springer)
• Evans. Partial Differential Equations (AMS)
• Lecture Notes (see resources)

Fluids:

• Landau Lifshitz. Course of Theoretical Physics Volume 6 Fluid Mechanics (Pergamon Press)
• Bachelor. An Introduction to Fluid Dynamics (Cambridge University Press)
• Serrin. Mathematical Principles of Classical Fluid Dynamics (Springer)
• Glowinski. Numerical Methods for Fluids (North Holland)
• Gresho. Incompressible Flow and the Finite Element Method vol 1+2 (Wiley)

Plan:

Chapter 1: Reminder Numerical Methods

1.1 Methods of characteristics and Riemann problem
1.2 Finite Differences
1.3 Finite Elements
1.4 Outlook

Chapter 2: Continuum Mechanics Framework
Chapter 3: Phenomenology of Fluid Flows
Chapter 4: Student Problems